Radial length contraction = mass anisotropy?

[Vincentius]

Mach's principle is considered falls by many because mass anisotropy would not be consistent with the Huges-Drever experiments, which showed the isotropy of nuclear resonance. However, anisotropy appears as well in GR in the form of radial length contraction, which can be interpreted as increase of inertia in the radial direction (the observer at infinity can not tell the difference). Thus one would expect anisotropic behavior of mass in GR just the same as according to Mach's principle. The fact that GR time dilation is isotropic does not change this anisotropic aspect of GR. So why is it an issue within Mach's principle and not within GR?

I tried to find a resource on a GR harmonic oscillator, but I can not find a paper that answers this question really.

Thanks for your input!

 

[PeterDonis]

anisotropy appears as well in GR in the form of radial length contraction, which can be interpreted as increase of inertia in the radial direction (the observer at infinity can not tell the difference).

Can you give a reference for this? I'm not sure what you're referring to.

 

[Dale]

However, anisotropy appears as well in GR in the form of radial length contraction, which can be interpreted as increase of inertia in the radial direction

I echo Peter Donis' comment. This needs a mainstream scientific reference.

 

[bcrowell]

Hughes-Drever experiments don't test Mach's principle, they test Lorentz invariance. See Will, "The Confrontation between General Relativity and Experiment," section 2.1.2, http://relativity.livingreviews.org/Articles/lrr-2006-3/fulltext.html . There is also a WP article: http://en.wikipedia.org/wiki/Hughes–Drever_experiment As both of these sources explain, the original motivation had to do with Mach's principle, but that turned out to be wrong.

Mach's principle can hold while Lorentz invariance also holds, as in Brans-Dicke gravity. Mach's principle can fail while Lorentz invariance holds (GR). I think most Lorentz-violating test theories do not embody Mach's principle, since they're typically designed to be minimal variations on SR/GR. So I don't think there's any close logical relationship between the two. Cocconi and Salpeter were working in 1958, long before BD gravity had clarified these issues by providing a viable Machian test theory.

Mach's principle is considered falls by many because mass anisotropy would not be consistent with the Huges-Drever experiments, which showed the isotropy of nuclear resonance.

No, Mach's principle (as embodied in BD gravity) is really not viable due to solar system tests. If Mach's principle had been falsified by Hughes-Drever in 1960, then BD gravity would never have been a viable, Machian theory.

However, anisotropy appears as well in GR in the form of radial length contraction, which can be interpreted as increase of inertia in the radial direction (the observer at infinity can not tell the difference).

No, length contraction does not create anistropy. GR is locally equivalent to SR, which is an isotropic theory.

 

[Naty1]

bcrowell posts:

Hughes-Drever experiments don't test Mach's principle, ... There is also a WP article: http://en.wikipedia.org/wiki/Hughes%...ver_experiment As both of these sources explain, the original motivation had to do with Mach's principle, but that turned out to be wrong.

I did not think the Wiki article said that...can you explain where I am going wrong here:
Wikipedia:

Heuristic arguments led them to believe that any inertial anisotropy, if one existed, would be dominated by mass contributions from the center of our Milky Way galaxy.

Nonuniform distribution of matter thus would lead to anisotropy of inertia in different directions. ... Hughes... and... Drever independently conducted similar spectroscopic experiments to test Mach's principle... made magnetic resonance measurements ...if inertia has a directional dependence, a triplet or broadened resonance line should be observed. ... Neither Hughes nor Drever observed any frequency shift of the energy levels, and due to their experiments' high precision, the maximum anisotropy could be limited to 0.04 Hz = 10−25 GeV.

From that I concluded the experiment showed NO directional deviation...negating Mach's idea...I seem to be missing something...[again!]

 

[Vincentius]

My question is not if Mach's principle is right, or Brans-Dicke theory, but instead why GR would not show anisotropic behavior. Radial length contraction is definitely anisotropic, so direction of the device must affect harmonic oscillator motion, I suspect. I am looking for an article on this subject. Has anyone a reference?

 

[PeterDonis]

Radial length contraction is definitely anisotropic

What do you mean by "radial length contraction"?

 

[Dale]

Vincentius, if you find a reference about the radial length contraction anisotropy then send me a PM and I will reopen the thread.

 

[Dale]

Hi Everyone, as requested Vincentius sent a message to me with a reference describing what he meant by gravitational length contraction. Here are the references:

Here is a reference
http://books.google.nl/books?id=D-g...a=X&ei=6346UaXVApDb7AaxhYGIDA&ved=0CCYQ6AEwBA

I suppose you don't question the phenomenon itself. But if so, this what Einstein himself tells about it:

English translation p.196-197: eq. (71)

http://web.archive.org/web/20060829045130/http://www.Alberteinstein.info/gallery/gtext3.html

Please remember to keep the discussion within the forum rules, i.e. use mainstream scientific sources and avoid personal speculation.

Vincentius, the equation in the Google Books reference is isotropic. The equation in the second one I will need to go over in more detail, but it looks simply like the use of non-isotropic coordinates rather than any physical anisotropy.

 

[PeterDonis]

The equation in the second one I will need to go over in more detail, but it looks simply like the use of non-isotropic coordinates

This is the way I read it as well. He appears to be using what we now call Schwarzschild coordinates, in which the radial metric coefficient $g_{rr}$ is different from the tangential coefficients $g_{\theta \theta}$ and $g_{\phi \phi}$. So the coordinates are non-isotropic; but that doesn't mean the unit length "measuring rods" he talks about are non-isotropic. Indeed, if you read carefully, you can see that he *assumes* that the measuring rods themselves *are the same length whether they are oriented radially or tangentially; that's how he derives the formulas in terms of the metric coefficients.

However, there *is* something "physical" going on here; but it's not anything to do with the measuring rods. It's to do with the geometry of space itself; the geometry of space as seen by an observer who is at rest in a static gravitational field is non-Euclidean. Einstein makes this clear in the paragraph following equation (71a).

 

[PAllen]

I recall a thread on this maybe a year ago. There are a sprinkling of sources that discuss this. In that thread, in an attempt to clarify a measurable versus coordinate effect, I proposed the idea of rulers laid out from a planet viewed orthogonally to the rulers at distance. Would the rulers closer to the planet subtend less angle than spatially flat model would predict? Unfortunately, neither I nor other participants at the time were willing to do the somewhat laborious computations needed to answer that question.

Here are more links that supporting the possibility of this idea:

http://www.mth.uct.ac.za/omei/gr/chap8/node8.html

http://mathpages.com/rr/s6-01/6-01.htm (search for length contraction)

 

[bcrowell]

I did not think the Wiki article said that...can you explain where I am going wrong here:

From that I concluded the experiment showed NO directional deviation...negating Mach's idea...I seem to be missing something...[again!]

Look at the rest of the article where they explain that that original interpetation was wrong.

 

[PAllen]

An additional comment on detecting length contraction: No observer detects anything but isotropy in their own local frame; this is obviously true even of the SR length contraction. It is a different observer who detects anisotropy in the affected object. Similarly for the hypothetical radial length contraction due to gravity - if it is observable it would be by distant observer as outlined in my scenario.

 

[Naty1]

Ben,or anybody, for the life of me I cannot see where the Hughes-Drever experiments are said to fail to test Mach's principle...This seems ok to me:
Vincentius posted:

Mach's principle is considered falls [failed??] by many because mass anisotropy would not be consistent with the Huges-Drever experiments, which showed the isotropy of nuclear resonance.

to clarify, I am not questioning isotropy in SR/GR...just how the Hughes-Drever experiments failed to test Mach's principle...

These Hughes-Drever experiments showed the isotropy of nuclear resonance to a very accurate tolerance, right?? In other words, they found no physical relationship between the the distant stars and a local local inertial frame as Mach would have predicted.

I had never heard of these experiments before and so am interested if they really DO negate Mach's principle...apparently not, but I am stumped why not.


You posted

As both of these sources explain, the original motivation had to do with Mach's principle, but that turned out to be wrong.

It's after the "but..." that I don't understand.

As I read the Wikipedia article, under MODERN INTERPRETATION, it says to me that after the Hughes-Drever experiments, it was realized the experiments ALSO applied to SR/GR:

While the motivation for this experiment was to test Mach's principle, it has since become recognized as an important test of Lorentz invariance and thus special relativity. This is because anisotropy effects also occur in the presence of a preferred and Lorentz-violating frame of reference – usually identified with the CMBR-rest frame as some sort of luminiferous aether (relative velocity ca. 368 km/s). Therefore, the negative results of the Hughes–Drever experiments (as well as the Michelson–Morley experiments) rule out the existence of such a frame. In addition, a fundamental consequence of the equivalence principle of general relativity is that Lorentz invariance locally holds in freely moving reference frames = local Lorentz invariance (LLI). This means that the results of this experiment concern both special and general relativity.

thanks...

 

[Vincentius]

I agree that locally everything is isotropic. And I also agree that proper spacetime near a mass is anisotropic in the eyes of a remote observer. Furthermore, relativistic trajectories like the perihelion precession are explained entirely by the anisotropy of the metric near a mass, so it is physical, not a coordinate artefact.
Then, if we have two identical harmonic oscillators on the same spot, one cycling in the radial direction of the massive body, the other one cycling in a perpendicular direction, then the two will run synchronously because of local isotropy (=Hughes-Drever). The remote observer, however, should be able to detect anisotropy in the motion of the oscillators, physically. But obviously, he still must see synchronous oscillators.
So, the question is: while anisotropy somehow affects the oscillators physically, why is it not visible in the cycle time? Hence, anisotropy either does not affect cycle time, or the effect is canceled by something else. This is what I like to get clearified. I would expect this has been explained times and over again, but I can not find a reference. Anyone knows? I appreciate your contribution.

 

[Dale]

I agree that locally everything is isotropic.

Excellent.

And I also agree that proper spacetime near a mass is anisotropic in the eyes of a remote observer.

Huh? How can you "agree that X" when nobody has said X. Furthermore, "proper spacetime" is not a standard term. I don't know what you are talking about. If you wish to discuss "proper spacetime" please provide a reference.

Furthermore, relativistic trajectories like the perihelion precession are explained entirely by the anisotropy of the metric near a mass, so it is physical, not a coordinate artefact.

No. You can use isotropic coordinates and get the exact same precessing geodesics:

http://en.wikipedia.org/wiki/Schwar...c.29_formulations_of_the_Schwarzschild_metric

So anisotropy in the Schwarzschild metric is a coordinate artifact and is not necessary to explain the geodesics.

The remote observer, however, should be able to detect anisotropy in the motion of the oscillators, physically.

How?

So, the question is: while anisotropy somehow affects the oscillators physically, why is it not visible in the cycle time?

We certainly haven't established that it does affect oscillators anisotropically in any coordinate-indepentent fashion.

 

[bcrowell]

Ben,or anybody, for the life of me I cannot see where the Hughes-Drever experiments are said to fail to test Mach's principle...As I read the Wikipedia article, under MODERN INTERPRETATION, it says to me that after the Hughes-Drever experiments, it was realized the experiments ALSO applied to SR/GR

Regardless of whether WP says so explicitly, the original interpretation was wrong, for the reasons I gave in #4.

 

[Vincentius]

Quote by Vincentius View Post

And I also agree that proper spacetime near a mass is anisotropic in the eyes of a remote observer.

Huh? How can you "agree that X" when nobody has said X. Furthermore, "proper spacetime" is not a standard term. I don't know what you are talking about. If you wish to discuss "proper spacetime" please provide a reference.

I agreed with PAllen #13. And I agree with you that "proper spacetime" is confusing. Just delete "proper".

Quote by Vincentius View Post

Furthermore, relativistic trajectories like the perihelion precession are explained entirely by the anisotropy of the metric near a mass, so it is physical, not a coordinate artefact.

No. You can use isotropic coordinates and get the exact same precessing geodesics:

http://en.wikipedia.org/wiki/Schwarz...zschild_metric

So anisotropy in the Schwarzschild metric is a coordinate artifact and is not necessary to explain the geodesics.

I can not imagine that one can derive perihelion precession from isotropic coordinates (that is, without converting back to anisotropic coordinates of course). Can you give a reference DaleSpam?

 

[PAllen]

I can not imagine that one can derive perihelion precession from isotropic coordinates (that is, without converting back to anisotropic coordinates of course). Can you give a reference DaleSpam?

It is essentially self evident that you can. Perihelion advance is a feature of shift in the angular coordinate of closest approach. Between isotropic SC coordinates and regular SC coordinates, the angular coordinates don't change (neither does the time coordinate). Only the radial is scaled. Thus, the transform of an orbit exhibiting with perihelion advance to isotropic coordinates shows exactly same advance, in exactly the same form. It remains a geodesic of the metric expressed in isotropic coordinates.

 

[Dale]

I can not imagine that one can derive perihelion precession from isotropic coordinates (that is, without converting back to anisotropic coordinates of course). Can you give a reference DaleSpam?

Sure. See Carrol's lecture notes, p 55-58 derives the covariant derivative to "perform the functions of the partial derivative, but in a way independent of coordinates". Then equation 3.30 and 3.46 define a geodesic in terms of the covariant derivative.

http://arxiv.org/abs/gr-qc/9712019

Since a geodesic is defined in terms of the covariant derivative, and since the covariant derivative is independent of coordinates, then if a path is a geodesic in anisotropic coordinates it remains a geodesic in any other coordinates, including isotropic coordinates. I.e. the orbit is the same orbit regardless of the coordinates.

 

[PAllen]

[Edit: I had this backwards from a sign error -1/2 exponent instead of 1/2 exponent. Corrected from original.]

There is a clear physical meaning (that Peter Donis #10 hinted at) for the quantity that some authors are calling radial length contraction. I will describe this explicitly and show:

- how you interpret this quantity is subject to definitions you choose
- there is some basis ot call the effect length contraction
- the result of my proposed distant observation (see post #11) should either show length contraction or the null result. Without doing the laborious calculation, I am still not sure.

First, the quantity computed e.g. in the first reference I give in my post #11 really has the following meaning describing a deviation from Euclidean spatial geometry:

If you take a ruler and measure the circumference of two concentric circles around an isolated, spherical, massive body; then compute the expected distance between them computing dividing each by 2π, and subtracting; then measuring the distance between them with the same ruler: you fill find the distance between measured this way is greater than expected by the factor computed in the reference.

Should this reasonably be called length contraction? Well, if you assume the tangential ruler is 'normal', then the radial ruler must shorten to show too large a reading between concentric circles (However, if you adopt simply non-euclidean geometry as the interpretation, then no rulers change size, by definition). If you assume the radial ruler is normal, then the tangentially oriented rulers are too long (thus measuring a circumference too short for the radial change). It is thus a matter of convention which ruler you might consider to change. It is more reasonable to avoid this arbitrary choice and describe the invariant feature that the ratio of radial difference to circumferential distance is greater than expected, indicating non-euclidean geometry.

As for my experiment in post #11, if it comes out with a null result - no change in subtended angle compared to flat space expectation - that favors the interpretation growth of tangential rulers. If it comes out showing the ruler subtending a smaller than expected angle, it supports interpretation that tangential rulers are unchanged and radial rulers shrink.

 

[Naty1]

Bcrowell...thanks for the confirmation in your post #17...

My problem had been that I don't really understand the foundations explained in your post #4 yet and since you said Hughes-Drever experiments fail to test Mach's principle as explained in Wikipedia, I did not know how to interpret the Wikipedia explanations...

Well, upon my fourth or so reading of Wikipedia at the end of this section

Experiments by Hughes and Drever

I came across this...

...Thus the null result is rather showing that inertial anisotropy effects are, if they exist, universal for all particles and locally unobservable.

And for those who may be interested,

this 'locally unobservable' statement seems to me to be confirmed in bcrowell's other reference

[ http://relativity.livingreviews.org/Articles/lrr-2006-3/fulltext.html ]

section 2.1...

 

[PeterDonis]

If you take a ruler and measure the circumference of two concentric circles around an isolated, spherical, massive body; then compute the expected distance between them computing dividing each by 2π, and subtracting; then measuring the distance between them with the same ruler: you fill find the distance between measured this way is less than expected by the factor computed in the reference.

You've got it backwards; the "ruler distance" between two concentric circles as seen by static observers in Schwarzschild spacetime is *larger* than expected if you divide each circumference by 2 pi and take the difference between them. The difference between the two circumferences (divided by 2 pi) is just the difference in the Schwarzschild radial coordinate; but this difference is *less* than the actual physical distance, because the actual physical distance is the difference in Schwarzschild r times the metric coefficient g_rr (more precisely, the square root of g_rr), and g_rr is greater than 1.

I agree that the term "radial length contraction" is not a good term for this phenomenon; but it comes about because people can't stop thinking about the Schwarzschild radial coordinate r as a physical "distance" instead of just a coordinate.

 

[PAllen]

You've got it backwards; the "ruler distance" between two concentric circles as seen by static observers in Schwarzschild spacetime is *larger* than expected if you divide each circumference by 2 pi and take the difference between them. The difference between the two circumferences (divided by 2 pi) is just the difference in the Schwarzschild radial coordinate; but this difference is *less* than the actual physical distance, because the actual physical distance is the difference in Schwarzschild r times the metric coefficient g_rr (more precisely, the square root of g_rr), and g_rr is greater than 1.

I agree that the term "radial length contraction" is not a good term for this phenomenon; but it comes about because people can't stop thinking about the Schwarzschild radial coordinate r as a physical "distance" instead of just a coordinate.

oops, you're right. I'll add a correction to my prior post.

 

[Vincentius]

Thanks for pointing out DaleSpam and PAllen. I agree that choice of coords does not matter and I admit that your comments made me rethink my earlier statement that isotropic coords can not explain the perihelion precession. The (isotropic) Eddington coords can. So we agree on that. Still I maintain that anisotropy in itself is indispensable in explaining the precession. So we may have a issue with terminology: The Schwarzschild metric in Eddington's coords reads

$c^2 {d \tau}^{2} = \frac{(1-\frac{r_s}{4r_1})^{2}}{(1+\frac{r_s}{4r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{r_s}{4r_1}\right)^{4}\left(dr_1^2 + r_1^2 d\theta^2 + r_1^2 \sin^2\theta \, d\varphi^2\right) \,.$

The coords may be called isotropic, the metric isn't, only locally. Ones you start integrating, the leading factor $\left(1+\frac{r_s}{4r_1}\right)^{4}$ changes value depending on the direction of motion (=anisotropy). Hence, radial and transverse directions are not symmetrical in the integral (that is, provided r changes value, as in elliptical orbit), which is IMO what anisotropy is about. This makes the term isotropic coords confussing. And surely, the Schwarzschild metric is anisotropic for a physical reason (getting closer to a mass or not), which can not be (and isn't) annihilated by a (valid) transform to Eddington coords. Or can you imagine that the anisotropy in the Schwarzschild metric doesn't matter at all in the integral?

 

[bcrowell]

Still I maintain that anisotropy in itself is indispensable in explaining the precession.

As far as I can tell, you've never defined what you mean by anisotropy, or if you did, you gave definitions that didn't make sense or weren't suitable. You seemed to have had in mind something coordinate-dependent, but nothing coordinate-dependent is ever real or meaningful in GR. The standard, coordinate-independent definition of isotropy is given here: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html . In somewhat less technical terminology than Carroll's, basically the definition is that at a certain point in space, the Riemann tensor takes on equal values in all 2-planes. By this definition, cosmological spacetimes can be isotropic, but the Schwarzschild metric is not. Basically I think you need to back up and learn a little more about the foundations of GR in order to have enough background to get anywhere on this. If you want to tell us a little about your background in math and physics, we could point you to some reading material.

 

[PAllen]

Thanks for pointing out DaleSpam and PAllen. I agree that choice of coords does not matter and I admit that your comments made me rethink my earlier statement that isotropic coords can not explain the perihelion precession. The (isotropic) Eddington coords can. So we agree on that. Still I maintain that anisotropy in itself is indispensable in explaining the precession. So we may have a issue with terminology: The Schwarzschild metric in Eddington's coords reads

$c^2 {d \tau}^{2} = \frac{(1-\frac{r_s}{4r_1})^{2}}{(1+\frac{r_s}{4r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{r_s}{4r_1}\right)^{4}\left(dr_1^2 + r_1^2 d\theta^2 + r_1^2 \sin^2\theta \, d\varphi^2\right) \,.$

The coords may be called isotropic, the metric isn't, only locally. Ones you start integrating, the leading factor $\left(1+\frac{r_s}{4r_1}\right)^{4}$ changes value depending on the direction of motion (=anisotropy). Hence, radial and transverse directions are not symmetrical in the integral (that is, provided r changes value, as in elliptical orbit), which is IMO what anisotropy is about. This makes the term isotropic coords confussing. And surely, the Schwarzschild metric is anisotropic for a physical reason (getting closer to a mass or not), which can not be (and isn't) annihilated by a (valid) transform to Eddington coords. Or can you imagine that the anisotropy in the Schwarzschild metric doesn't matter at all in the integral?

It is true that physically, toward, away, and tangential directions are distinguished around a massive body. It is true that the spatial metric components are constant for fixed r1 (changing angle), while they change for varying r1. They still have to express the same non-Euclidean spatial geometry of a static slice (a slice 4-orthogonal to the timelike killing vector field). As for attributing perihelion advance purely to the geometry of a spatial slice, I'm not so sure. There are coordinates for SC geometry where the spatial slices are Euclidean flat: Gullstrand–Painlevé coordinates. Clearly, space-time curvature causes the non-circular geodesics not to be closed, but it is not clear that thus must be attributed to spatial anisotropy. Evidence for this is that Newtonian gravity distinguishes the radial direction, yet has closed free fall elliptical paths.

 

[Dale]

Thanks for pointing out DaleSpam and PAllen. I agree that choice of coords does not matter and I admit that your comments made me rethink my earlier statement that isotropic coords can not explain the perihelion precession. The (isotropic) Eddington coords can. So we agree on that.

Excellent!

I maintain that anisotropy in itself is indispensable in explaining the precession. So we may have a issue with terminology: The Schwarzschild metric in Eddington's coords reads

$c^2 {d \tau}^{2} = \frac{(1-\frac{r_s}{4r_1})^{2}}{(1+\frac{r_s}{4r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{r_s}{4r_1}\right)^{4}\left(dr_1^2 + r_1^2 d\theta^2 + r_1^2 \sin^2\theta \, d\varphi^2\right) \,.$

The coords may be called isotropic, the metric isn't, only locally

So locally it is isotropic, and globally it is spherically symmetric, which is also considered isotropic.

Ones you start integrating, the leading factor $\left(1+\frac{r_s}{4r_1}\right)^{4}$ changes value depending on the direction of motion (=anisotropy). Hence, radial and transverse directions are not symmetrical in the integral (that is, provided r changes value, as in elliptical orbit), which is IMO what anisotropy is about

So, if you consider a patch of spacetime which is too large to consider "local" (I.e. Curvature effects are noticeable) but not so large as to contain the center, then I agree that you can have anisotropy in that patch. This anisotropy is not a feature of GR, per se, but a feature of the chosen boundary conditions. I.e. You have chosen to analyze a scenario where there is a big mass nearby in one direction, and not in any other.

Or can you imagine that the anisotropy in the Schwarzschild metric doesn't matter at all in the integral?

If you are using the isotropic coordinates then nothing about the usual Schwarzschild coordinates matters at all. The standard coordinates are not necessary in any way.